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<p>Suppose that <span class="process-math">\(f(x, y)\)</span> and <span class="process-math">\(\frac{\partial f}{\partial y}\)</span> are continuous in some rectangle <span class="process-math">\(\alpha \leq x \leq \beta\text{,}\)</span> <span class="process-math">\(\gamma \leq y \leq \delta\text{,}\)</span> which contains <span class="process-math">\((x_0, y_0)\text{.}\)</span> Then (<a href="" class="xref" data-knowl="./knowl/eq2_16.html" title="Equation 2.4.1">(2.4.1)</a>) has a unique solution which is valid in some interval <span class="process-math">\(x_0 -h \leq x \leq x_0+h\)</span> within <span class="process-math">\(\alpha \leq x \leq \beta\text{.}\)</span> <dfn class="terminology">Remark:</dfn> Notice the difference of conditions between linear and nonlinear ODEs. Here, the exact values <span class="process-math">\(h\)</span> is not stated in the theorem. It depends on the differential equation as well as the initial condition.</p>
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